Properties

Label 6042.2.a.t
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17609.1
Defining polynomial: \(x^{4} - x^{3} - 7 x^{2} + 10 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( 2 - \beta_{3} ) q^{5} - q^{6} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + ( 2 - \beta_{3} ) q^{5} - q^{6} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + q^{8} + q^{9} + ( 2 - \beta_{3} ) q^{10} + ( -\beta_{1} - \beta_{2} ) q^{11} - q^{12} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{13} + ( 1 + \beta_{2} - \beta_{3} ) q^{14} + ( -2 + \beta_{3} ) q^{15} + q^{16} + ( 2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{17} + q^{18} - q^{19} + ( 2 - \beta_{3} ) q^{20} + ( -1 - \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{2} ) q^{22} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} - q^{24} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{25} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{26} - q^{27} + ( 1 + \beta_{2} - \beta_{3} ) q^{28} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{29} + ( -2 + \beta_{3} ) q^{30} + ( -1 + \beta_{1} + \beta_{3} ) q^{31} + q^{32} + ( \beta_{1} + \beta_{2} ) q^{33} + ( 2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{34} + ( 5 - 3 \beta_{1} + \beta_{2} ) q^{35} + q^{36} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} - q^{38} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{39} + ( 2 - \beta_{3} ) q^{40} + ( 2 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{41} + ( -1 - \beta_{2} + \beta_{3} ) q^{42} + ( 5 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{44} + ( 2 - \beta_{3} ) q^{45} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{46} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{47} - q^{48} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{49} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{50} + ( -2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{51} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{52} - q^{53} - q^{54} + ( 2 - \beta_{2} - \beta_{3} ) q^{55} + ( 1 + \beta_{2} - \beta_{3} ) q^{56} + q^{57} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{58} + ( -4 + \beta_{1} ) q^{59} + ( -2 + \beta_{3} ) q^{60} + ( 1 - 2 \beta_{2} - 3 \beta_{3} ) q^{61} + ( -1 + \beta_{1} + \beta_{3} ) q^{62} + ( 1 + \beta_{2} - \beta_{3} ) q^{63} + q^{64} + ( -5 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{65} + ( \beta_{1} + \beta_{2} ) q^{66} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{68} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{69} + ( 5 - 3 \beta_{1} + \beta_{2} ) q^{70} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{71} + q^{72} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{73} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{74} + ( -3 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{75} - q^{76} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{77} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{78} + ( 3 + \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{79} + ( 2 - \beta_{3} ) q^{80} + q^{81} + ( 2 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{82} + ( -\beta_{2} + 2 \beta_{3} ) q^{83} + ( -1 - \beta_{2} + \beta_{3} ) q^{84} + ( 8 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{85} + ( 5 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{86} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{87} + ( -\beta_{1} - \beta_{2} ) q^{88} + ( 4 - \beta_{2} + \beta_{3} ) q^{89} + ( 2 - \beta_{3} ) q^{90} + ( 7 + \beta_{1} + \beta_{2} ) q^{91} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{92} + ( 1 - \beta_{1} - \beta_{3} ) q^{93} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{94} + ( -2 + \beta_{3} ) q^{95} - q^{96} + ( 5 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{97} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{98} + ( -\beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 6q^{5} - 4q^{6} + 3q^{7} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 6q^{5} - 4q^{6} + 3q^{7} + 4q^{8} + 4q^{9} + 6q^{10} - 2q^{11} - 4q^{12} - q^{13} + 3q^{14} - 6q^{15} + 4q^{16} + 8q^{17} + 4q^{18} - 4q^{19} + 6q^{20} - 3q^{21} - 2q^{22} + 12q^{23} - 4q^{24} + 6q^{25} - q^{26} - 4q^{27} + 3q^{28} - 4q^{29} - 6q^{30} - q^{31} + 4q^{32} + 2q^{33} + 8q^{34} + 18q^{35} + 4q^{36} + 9q^{37} - 4q^{38} + q^{39} + 6q^{40} + 6q^{41} - 3q^{42} + 12q^{43} - 2q^{44} + 6q^{45} + 12q^{46} + 3q^{47} - 4q^{48} + 9q^{49} + 6q^{50} - 8q^{51} - q^{52} - 4q^{53} - 4q^{54} + 5q^{55} + 3q^{56} + 4q^{57} - 4q^{58} - 15q^{59} - 6q^{60} - 4q^{61} - q^{62} + 3q^{63} + 4q^{64} - 13q^{65} + 2q^{66} + 15q^{67} + 8q^{68} - 12q^{69} + 18q^{70} + 4q^{72} + 11q^{73} + 9q^{74} - 6q^{75} - 4q^{76} - 11q^{77} + q^{78} + 8q^{79} + 6q^{80} + 4q^{81} + 6q^{82} + 3q^{83} - 3q^{84} + 29q^{85} + 12q^{86} + 4q^{87} - 2q^{88} + 17q^{89} + 6q^{90} + 30q^{91} + 12q^{92} + q^{93} + 3q^{94} - 6q^{95} - 4q^{96} + 16q^{97} + 9q^{98} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 7 x^{2} + 10 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + \nu^{2} - 6 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1} - 4\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.108328
2.21368
−2.80740
1.48539
1.00000 −1.00000 1.00000 −1.35130 −1.00000 −2.98827 1.00000 1.00000 −1.35130
1.2 1.00000 −1.00000 1.00000 0.434198 −1.00000 1.90038 1.00000 1.00000 0.434198
1.3 1.00000 −1.00000 1.00000 3.28211 −1.00000 4.88149 1.00000 1.00000 3.28211
1.4 1.00000 −1.00000 1.00000 3.63500 −1.00000 −0.793614 1.00000 1.00000 3.63500
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6042.2.a.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.t 4 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(1\)
\(53\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{4} - 6 T_{5}^{3} + 5 T_{5}^{2} + 15 T_{5} - 7 \)
\( T_{7}^{4} - 3 T_{7}^{3} - 14 T_{7}^{2} + 19 T_{7} + 22 \)
\( T_{11}^{4} + 2 T_{11}^{3} - 11 T_{11}^{2} + 7 T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 1 - 6 T + 25 T^{2} - 75 T^{3} + 193 T^{4} - 375 T^{5} + 625 T^{6} - 750 T^{7} + 625 T^{8} \)
$7$ \( 1 - 3 T + 14 T^{2} - 44 T^{3} + 120 T^{4} - 308 T^{5} + 686 T^{6} - 1029 T^{7} + 2401 T^{8} \)
$11$ \( 1 + 2 T + 33 T^{2} + 73 T^{3} + 483 T^{4} + 803 T^{5} + 3993 T^{6} + 2662 T^{7} + 14641 T^{8} \)
$13$ \( 1 + T + 10 T^{2} + 8 T^{3} + 40 T^{4} + 104 T^{5} + 1690 T^{6} + 2197 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 8 T + 35 T^{2} - 183 T^{3} + 929 T^{4} - 3111 T^{5} + 10115 T^{6} - 39304 T^{7} + 83521 T^{8} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( 1 - 12 T + 133 T^{2} - 849 T^{3} + 5001 T^{4} - 19527 T^{5} + 70357 T^{6} - 146004 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 4 T + 93 T^{2} + 291 T^{3} + 3860 T^{4} + 8439 T^{5} + 78213 T^{6} + 97556 T^{7} + 707281 T^{8} \)
$31$ \( 1 + T + 104 T^{2} + 105 T^{3} + 4566 T^{4} + 3255 T^{5} + 99944 T^{6} + 29791 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 9 T + 130 T^{2} - 884 T^{3} + 7016 T^{4} - 32708 T^{5} + 177970 T^{6} - 455877 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 6 T + 129 T^{2} - 553 T^{3} + 7352 T^{4} - 22673 T^{5} + 216849 T^{6} - 413526 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 12 T + 81 T^{2} - 249 T^{3} + 315 T^{4} - 10707 T^{5} + 149769 T^{6} - 954084 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 3 T + 89 T^{2} - 180 T^{3} + 4758 T^{4} - 8460 T^{5} + 196601 T^{6} - 311469 T^{7} + 4879681 T^{8} \)
$53$ \( ( 1 + T )^{4} \)
$59$ \( 1 + 15 T + 313 T^{2} + 2817 T^{3} + 30091 T^{4} + 166203 T^{5} + 1089553 T^{6} + 3080685 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 4 T + 103 T^{2} - 175 T^{3} + 4822 T^{4} - 10675 T^{5} + 383263 T^{6} + 907924 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 15 T + 252 T^{2} - 2148 T^{3} + 22478 T^{4} - 143916 T^{5} + 1131228 T^{6} - 4511445 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 232 T^{2} + 72 T^{3} + 22926 T^{4} + 5112 T^{5} + 1169512 T^{6} + 25411681 T^{8} \)
$73$ \( 1 - 11 T + 314 T^{2} - 2344 T^{3} + 35050 T^{4} - 171112 T^{5} + 1673306 T^{6} - 4279187 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 8 T + 275 T^{2} - 1429 T^{3} + 30073 T^{4} - 112891 T^{5} + 1716275 T^{6} - 3944312 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 3 T + 296 T^{2} - 716 T^{3} + 35354 T^{4} - 59428 T^{5} + 2039144 T^{6} - 1715361 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 17 T + 447 T^{2} - 4693 T^{3} + 63741 T^{4} - 417677 T^{5} + 3540687 T^{6} - 11984473 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 16 T + 337 T^{2} - 4355 T^{3} + 47269 T^{4} - 422435 T^{5} + 3170833 T^{6} - 14602768 T^{7} + 88529281 T^{8} \)
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